On the Euclidean space and more generally on Riemannian manifolds, when the cost \(c\)  is the distance squared denoted \(d^2\) , a dynamic formulation of the Kantorovich's transport was first introduced in [1]}. This so-called Benamou-Brenier formulation consists in the minimization of the kinetic energy of the probability measure under the constraint of the continuity equation. Let us denote \(\mu (t,x)\)  a time-dependent density on the domain \(X\)  assumed to be a closed convex domain in \(^d\)  or a closed Riemannian manifold. This density follows the continuity equation which reads, for \(t \in [0,1]\)  and \(x \in X\)
\(\partial _t \mu (t,x) + \nabla \cdot (\mu (t,x) v(t,x)) = 0\,,\) 
